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Final Examination December 2005 - Mathematics 400, Exams of Mathematics

Aliah UniversityMathematics

The final examination questions for mathematics 400 at the university of british columbia, december 2005. The examination covers nonlinear pdes, polar coordinates, heat conduction, and the telegraph equation. Students are required to find solutions, provide plots, and determine eigenvalues and eigenfunctions.

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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The University of British Columbia
Final Examination - December 9, 2005
Mathematics 400
Instructor: Dr. A. Cheviakov
Closed book examination Time: 2.5 hours
Name Signature
Student Number Section 101 102 (circle one)
Special Instructions:
- Be sure that this examination has 7 pages. Write your name on top of each page.
- Submit only this booklet, with solution written in space provided (you may use adjacent page(s)).
Clearly outline answers. Solutions on scratch paper will not be graded.
- A 2-sided self-prepared Letter-size formula sheet is allowed. No calculators or notes are permitted.
- In case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit
quickly and quietly to a pre-designated location.
Rules governing examinations
•Each candidate should be prepared to produce her/his
library/AMS card upon request.
•No candidate shall be permitted to enter the examination
room after the expiration of one half hour, or to leave during
the first half hour of examination.
•Candidates are not permitted to ask questions of the in-
vigilators, except in cases of supposed errors or ambiguities
in examination questions.
•CAUTION - Candidates guilty of any of the following or
similar practices shall be immediately dismissed from the
examination and shall be liable to disciplinary action.
(a) Making use of any books, papers, or memoranda, other
than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other
candidates.
•Smoking is not permitted during examinations.
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2 20
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5 20
Total 100
Page 1 of 7
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Download Final Examination December 2005 - Mathematics 400 and more Exams Mathematics in PDF only on Docsity!

The University of British Columbia Final Examination - December 9, 2005 Mathematics 400 Instructor: Dr. A. Cheviakov

Closed book examination Time: 2.5 hours

Name Signature

Student Number Section 101 102 (circle one)

Special Instructions:

  • Be sure that this examination has 7 pages. Write your name on top of each page.
  • Submit only this booklet, with solution written in space provided (you may use adjacent page(s)). Clearly outline answers. Solutions on scratch paper will not be graded.
  • A 2-sided self-prepared Letter-size formula sheet is allowed. No calculators or notes are permitted.
  • In case of an exam disruption such as a fire alarm, leave the exam papers in the room and exit quickly and quietly to a pre-designated location.

Rules governing examinations

  • Each candidate should be prepared to produce her/his library/AMS card upon request.
  • No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of examination.
  • Candidates are not permitted to ask questions of the in- vigilators, except in cases of supposed errors or ambiguities in examination questions.
  • CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers, or memoranda, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates.
  • Smoking is not permitted during examinations.

Total 100

Page 1 of 7

[15 pts] Problem 1. Find the solution y(x, t) of the nonlinear PDE problem

yt + (1 − 3 y^2 )yx = 0 − ∞ < x < +∞, t > 0

y(x, 0) =

2 / 3 , x < 0; 0 , x > 0.

Provide a plot showing characteristics.

[20 pts] Problem 3. Consider heat conduction problem in a 2D infinite strip of width H.

ut = k(uxx + uyy), 0 < y < H, − ∞ < x < +∞, t > 0

uy(x, 0 , t) = uy(x, H, t) = 0, u(x, y, 0) = f (x, y).

Here u(x, y, t) is temperature, and f (x, y) a smooth function absolutely integrable in the strip. (i) Solve the problem to find u(x, y, t) for t > 0. (ii) Find the equilibrium heat distribution as t → ∞. (iii) Find the exact form of the solution, if f (x, y) = e−x 2 sin(2πy/H).

[25 pts] Problem 4. Consider the Linear Telegraph Equation problem:

uxx(x, t) = αutt(x, t) + βut(x, t) + γu(x, t), x > 0 , t > 0

u(0, t) = f (t), u(x, 0) = ut(x, 0) = 0.

Here α, β, γ ≥ 0 are constants. This PDE problem describes electromagnetic signal transmission along a semi-infinite cable; x is a coordinate along the cable, and t ≥ 0 time. u(x, t) is bounded for all x, t > 0.

(i) Find the Laplace transform (in time) U (x, s) of the solution u(x, t). (ii) Suppose β^2 = 4αγ. Find the solution u(x, t). Write it in the form that explicitly contains necessary Heaviside function(s). (iii) Verify by substitution that your solution satisfies the initial-boundary value problem. (iv) Specify another set of constants α, β, γ (β^2 6 = 4αγ), for which the solution u(x, t) can be explicitly found. Find it.